My Truck needs some Tums

Tonight, Sarah was going to the grocery store. My truck was behind her car in the driveway so I told her to take mine. Well she went to pick up her purse that she had left at a friends house. Well when she got back out the truck would not start. She got a jump and made it to the grocery store where I met her to switch cars to take the truck. I got out to get in the truck... she cuts off and dies right there. I jump her again, make it 400 feet or so to the light at the end of the parking lot and have to put up onto the concrete median and wait for Sarah to get done in the store. Jump number three, I am on my way home and start to get closer to our house. The headlights start to dim as the interior lights join them. As I see the turn coming up down the hill, the truck dies again. I must remind you that the brakes really suck with it being off and the power steering goes bye bye. I muscle that bad boy into our neighborhood, run a stop sign to maintain speed and hope to make it up the hill, I don't. I hop out and begin to push and steer to try to get over the hill. Might I add that Maple is in shotgun during all of this looking at me. As I am pushing the truck, Maple thinks its time to play, maybe this is some kind of new game. So now I am battling her as well in addition to this 1/2 ton truck. I crested the hill and started towards our driveway, I made it almost all the way in... Jump 4... in the driveway and trying to figure out if it is the battery or the alternator.